\section{Antenna}
\label{appendix:antenna}

\subsection*{Efficiency loop antenna}
A loop antenna has 2 important resistances for the efficiency. The radiated resistance and the loss resistance. For optimal efficiency the radiated resistance has to be much larger then the loss resistance.  

The radiated resistance of a loop antenna can be calculated with this formula:
\begin{equation}
R_r=31.171N^2\frac{S^2}{\lambda^4}
\end{equation}
Where: \\
$a$ is the radius of the antenna.  \\
$S$ is the surface of the antenna and thus equal to $\pi{}a^2$.  \\
$N$ is the amount of loops. \\
$\lambda$ is the wavelength.  \\

The loss resistance of a loop antenna can be calculated with:
\begin{equation}
R_s=\frac{Na}{b}\sqrt{\frac{\omega\mu_0}{2\sigma}}(\frac{R_p}{R_0}+1) 
\end{equation}
Where: \\
$f$ is the frequency. \\
$\omega{}$ is $2\pi{}f$.  \\
$\sigma{}$ is the conductivity of the material, the conductivity of copper is: $5.96e7$ at $20 ^\circ C$.\\
$b$ is the radius of the wire antenna.\\
$\frac{R_p}{R_0}$ is the extra resistance factor due the proximity effect.  \\

Now with these two formulas the efficiency of a loop antenna can be calculated \cite{AntennaBalansis}. 
\begin{equation}
Eff=\frac{R_r}{R_r+R_s}
\end{equation}

The efficiency is dependent on the amount of windings and the radius of the loop antenna. Also the efficiency is dependent on the material and thickness. With more loops though the proximity effect $(\frac{R_p}{R_0})$ will play a role, and will decrease the efficiency. See \ref{appendix:proxeffect}.
In the following \cref{1 loop Antenna Efficiency} the efficiency of a 1 loop antenna is plotted over the radius. Only 1 loop is plotted because increasing the surface twice as much or adding 1 extra loop, will ideally be the same. Different wire thickness's are plotted though.

\begin{figure}[ht!]
	\centering
	\includegraphics[width=150mm]{assets/eff.png}
	\caption{1 loop antenna efficiency}
	\label{1 loop Antenna Efficiency}
\end{figure}

\subsection*{Proximity effect}
\label{appendix:proxeffect}
 
If a conductor carries alternating current, alternating magnetic fields are created. These magnetic fields influence the currents of nearby conductors. This is called the proximity effect, and will thus happen with 2 or more loops.
What will happen is that there occurs a current crowding due the magnetic fields. This crowding is concentrated in the areas of the conductor furthest away from nearby conductors carrying current in the same direction. So in theory the copper is not effectively used by the electrons. And this will thus increase the resistance. 
How strong the effect is, is depending on the space between the windings, and the diameter of the windings. If you look at \cref{Factor proximity effect} and \cref{Dimensions inductor} there will be a certain $(\frac{R_p}{R_0})$ \cite{AntennaBalansis} which is the factor of additional extra resistance due the proximity effect. Now as you can see this factor can become easily bigger then 1, and can’t be ignored under certain conditions.

\begin{figure}[ht!]
	\centering
	\includegraphics[width=90mm]{assets/prox.png}
	\caption{Factor proximity effect, \cite{AntennaBalansis}}
	\label{Factor proximity effect}
\end{figure}

\begin{figure}[ht!]
	\centering
	\includegraphics[width=40mm]{assets/ca.png}
	\caption{Dimensions inductor, \cite{AntennaBalansis}}
	\label{Dimensions inductor}
\end{figure}

\subsection*{Inductance loop antenna}

The inductance of a loop antenna in air can be estimated with the following formula \cite{NXPant}:
\begin{equation}
L(nH)= \frac{24.6 \cdot N_a^2 \cdot D^2[cm]}{D[cm]+2.75\cdot s[cm]} 
\end{equation}
Where:\\
$D$ is the diameter of the antenna.\\
$N_a$ is the amount of windings.\\
$s$ is the width of the antenna.\\

Now plots of this formula with certain amount of windings are made:
In the highlighted places the antennas with different amount of loops have the same efficiency, assuming its all made of the same material and ignoring the proximity effect. Now as you can see the inductance of 1 loop is much lower. Now of course the inductance the extra loops will add, is dependent on the amount of width between the loops. Smaller width, will add more inductance.

\begin{figure}[ht!]
	\centering
	\includegraphics[width=150mm]{assets/induc.png}
	\caption{Inductance loop antenna}
	\label{Inductance loop antenna}
\end{figure}

\subsection*{Resonance frequency and Quality factor}

A RLC circuit has a certain resonance frequency and quality factor. The resonance frequency for an loop antenna is extremely important, this can be the difference in no signal at all, and a relative large signal. Setting the resonance frequency can be done by adding a capacitor. So far it is thus an LC circuit.
The resonance frequency for a parallel LC circuit can be calculated with the following formula:
\begin{equation}
f_res=\frac{1}{2\pi{}(LC))}
\end{equation}
If $L = 1.25\mu{}H$, a parallel capacitance of around $110pF$ is needed for a resonance frequency of $13.56 MHz$. For $5\mu{}H$ this is $27.5pF$. And for $L = 10\mu{}H$ this is $13.77pF$.

Because the inductor value (L) is fixed, a certain parallel capacitance is added. 
The Quality factor is in fact the bandwidth of the antenna. Increasing the bandwidth can be done by adding a parallel resistance. The relation between the bandwidth and the RLC circuit values is:
\begin{equation}
Q=R\sqrt{\frac{C}{L}} = \frac{f_c}{f_2-f_1}
\end{equation}

\begin{figure}[ht!]
	\centering
	\includegraphics[width=100mm]{assets/bandwidth.png}
	\caption{Quality factor and Bandwidth, \cite{QualityFactorwiki}}
	\label{Quality factor and Bandwidth}
\end{figure}

\subsection*{Measurement Results}

Here are some measurement results of the antenna. This measurement was mainly to check if the signal indeed drops with $r^{-3}$. Now as can be seen in \cref{Measurement Antenna} it does indeed fall off with this rate. 

\begin{figure}[ht!]
	\centering
	\includegraphics[width=120mm]{assets/measurement^3.png}
	\caption{Measurement Antenna}
	\label{Measurement Antenna}
\end{figure}

In the following \cref{Spectrum antenna} the antenna is connected to the spectrum analyser.

\begin{figure}[ht!]
	\centering
	\includegraphics[width=120mm]{assets/spectrumantenna.png}
	\caption{Spectrum antenna}
	\label{Spectrum antenna}
\end{figure}